16 The purpose of this analysis has been to determine when branches of solu- tions are related by a simple scaling, since the numerical computation of two such branches is clearly unnecessary and wasteful. Thus, we conclude that it is sufficient to compute only those branches of solutions whose isotropy subgroup fl satisfies ei- ther r# = ~ or flo # fl and ~ is not a subgroup of ft. However, it may be necessary to scale the solution up (in the second case) in order to check for the existence of bifurcation points on the scaled branch which do not occur on the primary branch. Finally, we consider briefly the possibility that for a particular group P, there may be many scaling transformations/zk and constants bk, ok, Ik E R\{0} which sat- isfy (3.5) and (3.6) associated with epimorphisms ~k- We then have the following result which involves a monoid, a set which is closed under an associative binary operation and which has an identity element (ie. a group without inverses). Theorem 4.8 The sets B = {/~k} of epimorphisms on I ~ and H = {hk} of scaling functions which satisfy (3.5) and (3.6) have the structure of a monoid (with the operation of composition). Proof Since all the flk's are epimorphisms on F, the composition flkflj is also an epi- morphism on r and composition of homomorphisms is associative. The trivial homomorphism fll - Id ]r is clearly an epimorphism and satisfies fllflk = flkfll = flk for all flk E B. Thus fll is the identity element of the set B. Since the flk's are not required to be injective, only surjective, they will not have inverses in general. Th~ts, the set B has the structure of a monoid. Each of the scaling functions hk : X -~ X gk (where Zk = ker ilk) satisfies We define P'k,j = ker(flkflj). (Note that in general ~k,j # ~]j,k.) Then flj(~k,j) = ~]k

17 and from Lemma 4.2, hj : X ~k --~ X~h,~ is an orthogonaJ linear homeomorphism. Thus hjhk : X ~ X ~k,j is also an orthogonal linear homeomorphism which satisfies T(7)hihk = hjhkT(flkflj(7) ) and so the composition hjh~ is a va~d scaling transformation associated with the epimorphism flkflj. Also, composition of maps is associative. The trivial mapping hi = I Ix is an orthogonal linear homeomorphism on X which satisfies (3.5) with = fll and (3.6) with b = c = I = 1 and is thus the identity of the set H. Again there cannot be inverses for the hk's (other than hi) since their range is not the whole of X. Thus, the set H has the structure of a monoid. [] We note that the definition (3.5) of the scaling function h is a property relat- ing group representations on Hilbert spaces and is independent of the particular equation under consideration. Thus the monoid structure of the scaling functions depends on the group F and not the equation. 5. The Kuramoto-Sivashinsky Equation A steady-state version of the KS equation in one dimension is given by g(~, ~) - 4~('~) + ~(~" + ~') = 0. (5.1) We seek 2~r-periodic solutions of (5.1) which have zero mean and so we define H m to be the Hilbert space of 2~r-perlodic functions with zero mean whose (weak) derivatives up to and including the m *h are square integrable. We define inner products on these spaces by < ~,v >~= 7J0 ~(m)(*)V(m)(~)d*" (5.2) Then g : X x BL --* Y where X = H 4 and Y = H °. We define an action of the group 0(2) (generated by R~, a e [0, 2zt) and S) on Y by R~(~) = ~(~ + ~), ~ e [0,2~) s~(,) = -~(-~).